M.Sc Thesis

M.Sc StudentAbu Hamed Mohammad
SubjectFrobenius-Schur Indicators for the Representation Categories
of Semisimple Lie Algebras
DepartmentDepartment of Mathematics
Supervisor PROF. Shlomo Gelaki


Let V be a finite dimensional irreducible representation of a complex semisimple Lie algebra g and let m 2 be an integer. The m-th Frobenius-Schur indicator of V is the number , where  is the cyclic automorphism of .  In this thesis we derive an explicit formula for the Frobenius-Schur indicators of  V. Our formula is given in terms of the dimensions of the weight spaces of  V, the roots of g and the Weyl group. As a result we obtain that when m is large enough the Frobenius-Schur indicators of degree m are all equal to the dimension of the zero weight space of  V. Next we introduce some properties of the 2-nd Frobenius-Schur indicator of V which are analogous to the classical case of the representation category of a finite group. Namely we show that  if and only if V is not self dual and if V is self dual, then   if and only if V admits a nondegenerate bilinear symmetric g -invariant form and  if and only if V admits a nondegenerate bilinear skew-symmetric g -invariant form. In addition we prove Tits theorem which gives an explicit formula for . Finally we calculate the Frobenius-Schur indicators for the irreducible representations of . In particular we give explicit formulas for those irreducible representations whose highest weight belongs to the root lattice of g.